x x " Q, f(x) = -x x " R \ Q, R R f : X Y X Y A " X B " Y A f
f(A) = {y " Y : y = f(x)}. x"A B f
f-1(B) = {x " X : y = f(x)}. y"B f(x) = x2, x " R A1 = R, A2 = [0, + "), A3 = N A4 = Z B1 = (-2, -1) B2 = [1, + ") B3 = {n " N : n = k4, k " N} f(A1) = [0, ") f(A2) = [0, ") f(A3) = {n " N : n = k2, k " N} f(A4) = {0} *" f(A3) f-1(B1) = " f-1(B2) = (- ", 1] *" [1, ") f-1(B3) = Z \ {0} f : X Y A1, A2 " X B1, B2 " Y 1. f(A1 *" A2) = f(A1) *" f(A2) 2. f(A1 )" A2) " f(A1) )" f(A2) 3. f-1(B1 *" B2) = f-1 *" f-1(B2) 4. f-1(B1 )" B2) = f-1(B1) )" f-1(B2) 2. f(x) = x2, x " R A1 = [-1, 0] A2 = [0, 1] f(A1 )" A2) = f({0}) = {0}. f-1([-1, 0]) = [0, 1] = f-1([0, 1]) ! f-1(A1) )" f-1(A2) = [0, 1]. f-1(A1 )" A2) f-1(A1) )" f-1(A2) 2. f : X Y Y f(X) = Y f : Df R Df " R f : Df R Df " Rn n " N f {(x, y) " R2 : y = f(x)}. f(x) = ax + b, x " R, (a, b " R). a b f(x) = ax2 + bx + c, x " R, (a, b, c " R, a = 0).
n n
f(x) = akxn-k, x " R, (a0, . . . , an " R, a0 = 0), n " N.
k=0 ax + b d f(x) = , x " R \ {- }, (a, b, c, d " R, c = 0, ad - bc = 0).
cx + d c a y = c x = -d f c P (x) f(x) = , x " R \ {x " R : Q(x) = 0}, Q(x) P Q f(x) = ax, x " R, (a " (0, 1) *" (1, + ")). R (0, + ") fm(x) = axm, x " Df , (m " R). m
0 = a0, k=0 ak n , n = 0, 1, . . . . n+1 = ak + an+1, k=0 k=0
ad - bc = 0 A = a b Det(A) = ad - bc c d Df fm m m = 2 Df = R m 2 1 m = Df = [0, + ") 1/2 2 f(x) = logax, x " (0, + "), (a " (0, 1) *" (1, + ")). f(x) = sin x, x " R. f(x) = cos x, x " R. sin x Ą f(x) = tgx a" , x " R \ {x " R : x = + kĄ, k " Z}. cos x 2 cos x f(x) = ctgx a" , x " R \ {x " R : x = kĄ, k " Z}. sin x f : Df Y, g : Dg W f(Df) " Dg h = g ć% f : Df W df h(x) = (g ć% f)(x) = g(f(x)), x " Df, f g df x a (logax = b, b " R) ! ab = x a logaxp = p logax, p " R logaax = x, x " (0, + ") alog x = x, x " R loga(x y) = logax + logay, x, y > 0 " f(x) = x2, x " R, g(x) = x, x 0 f : R [0, + ") g : [0, + ") [0, + ") f(R) = [0, + ") = Dg " (g ć% f)(x) = g(f(x)) = g(x2) = x2 = |x|, x " R. g(Dg) = [0, + ") " Df = R f ć% g " " (f ć% g)(x) = f(g(x)) = f( x) = ( x)2 = x, x " [0, + "). f g " " f(x) = -x, x 0 g(x) = x, x " [0, + ") Dg = [0, + ") = f(Df) g ć% f
" " " 4 (g ć% f)(x) = g(f(x)) = g( -x) = -x = -x, x " (- ", 0]. f ć% g f(Dg) = [0, + ") " Df = (- ", 0] f : X Y g : Y X f
g(y) = x ! f(x) = y. y"Y f : X Y g : Y X
(g ć% f)(x) = x '" (f ć% g)(y) = y. x"X y"Y f g f f g (x, f(x)) f y = x, x " R (g(y), y) y = f(x) x " Df f(x) = x2, x " (- ", 0] Df = (- ", 0] Pf = [0, + ") g : [0, + ") (- ", 0] f " " " " g(y) = x ! f(x) = y ! x2 = y ! x2 = y ! |x| = y ! x = - y. x y g g : [0, + ") (- ", 0] " x g(x) = - x. " (g ć% f)(x) = g(f(x)) = g(x2) = - x2 = -|x| = -(-x) = x, x " (- ", 0]. " " (f ć% g)(x) = f(g(x)) = f(- x) = (- x)2 = x, x " [0, + "). f : Df Y " = D " Df
h = f|D : D Y x h(x) = f(x) f D f : R [-1, 1] f(x) = sin x x " Df = R D Ą Df D = [-Ą , ] h f D 2 2 Ą h = f|D : [-Ą , ] [-1, 1] 2 2 x h(x) = sin x Ą [-Ą , ] [-1, 1] h 2 2 h g h x " D g(x) arcsinx x Ą arcsin : [-1, 1] [-Ą , ] 2 2 x arcsinx. Ą Ą df (h ć% f)(x) = arcsin(sin x) = x, x " [- , ] 2 2 df (f ć% h)(x) = sin arcsinx = x, x " [-1, 1]. Ą f(x) = sin x, x " [-Ą , ] 2 2 arcsin [-1, 1] Ą sinus 0, 6 arcsin(0) = Ą 0, arcsin(1) = 2 6 f f = cos|[0,Ą] : [0, Ą] [-1, 1] df x f(x) = cos x. f [0, Ą] [-1, 1] g f arccos g = arccos : [-1, 1] [0, Ą] x arccosx. Ą arccos1 = 0, arccos0 = 2 Ą (") arcsinx + arccosx = , x " [0, 1]. 2 Ą u = arcsinx v = arccosx u, v " [0, ] 2 x = sin u = cos v Ą sin u = cos v ! sin u = sin ( - v) ! 2 Ą Ą u = - v + 2kĄ, k " Z (" u = Ą - ( - v) + 2lĄ, l " Z. 2 2 Ą Ą k = 0 u = - v u + v = 2 2 k = 0
Ą k = 0 u-v = 2 Ą u = v = 0 k 2 Ą u + v = (") 2 Ą f : (-Ą , ) R 2 2 df x f(x) = tgx. Ą Ą f (-Ą , ) (-Ą , ) 2 2 2 2 R g f arctg Ą g = arctg : R (-Ą , ) 2 2 x arctgx. arctg0 = 0, arctgĄ = 1 4 f : (0, Ą) R df x f(x) = ctgx. f (0, Ą) R g f g = arcctg g = arcctg : R (0, Ą) x arcctgx. Ą Ą arcctg1 = , arcctg0 = 4 2 A, B, C, . . . , X, Y, Z, . . . a A " a " A A a a A X Y X Y X Y = {(x, y) : x " X '" y " Y } (x, y) x X y Y X Y X Y X " X X X X d : X [0, + ") X
a) d(x, y) = d(y, x) x,y"X b) [d(x, y) = 0 ! x = y] x,y"X c) d(x, y) d(x, z) + d(z, y) x,y,z"X (X, d) a = b d(a, b) > 0
X = R d(x, y) = |x - y|, x, y " R R d : R R [0, + ") a), b), c) R x0 " X r > 0 df K0(x0, r) = {x " X : d(x0, x) < r}. R K0(x0, r) = (x0 - r, x0 + r) x0 r > 0 df K-(x0, r) = {x " X : d(x0, x) r}. R K-(x0, r) = [x0 - r, x0 + r] (X, d) A X
K0(x, r) " A. x"A K0(x,r) R A X \ A A X X R {x} A (X, d) K0(x0, r) X A " K0(x0, r) A (X, d)
A \ K0(x0, r) = ".
K0(x0,r)"X f a" (an)n"N : N X not n f(n) = an X f a1, a2, . . . (an)n"N (an) an, n " N (an) f = (an)n"N : N N g = (bn)n"N g = f ć% bn = a(n), n " N, (an) 1 1 R an = , n " N bn = , n " N n n2 (an)
an-1 n = 2k, k " N, cn = an+1 n = 1, 3, . . . , 2k - 1, k " N, (an) (X, d) (an) X a " X
an " K0(a, ). no"N n n0 K0(a, ) R
|an - a| < . >0 n n0 n0"N a (an) (X, d) lim an = a (" an n " a - n " n an a an a n an = a, n " N limn " an = a limn " an = a limn " an = b a = b
d(a.b) = > 0
d(an, a) < 2 >0 n n0 n0"N
d(an, b) < . 2 >0 n0 n n0 n max{n0, n0}
0 < = d(a, b) d(a, an) + d(an, b) < + = . 2 2 > 0 < X = R R (an)n"N
|an| M. M"R n"N R (an), (xn) (bn) R
10 an xn bn n0"N n n0 2o limn " an = x limn " bn = x (xn) limn " xn = x (xn) (yn) R x y
xn yn, n0"N n n0 x y (xn) R
(C) |xn - xn+m| < . >0 n0"N n n0,m"N R R (an) (bn) limn "(an ą bn) = limn " an ą limn " bn limn "(can) = c limn " an c " R limn " an bn) = (limn " an) (limn " bn) limn " an limn " an = limn " bn = 0
bn limn " bn limn "(an)p = (limn " an)p p " Z \ {0} " " k k limn " an = limn " an k " N \ {1} an x
|an - x| < . >0 n0"N n n1 bn x
|bn - x| < . >0 n0"N n Ć Ć n0 n H0 = max{n0, n0, n0} Ć x - < an < xn < bn < x + ,
|xn - x| < , n H0 xn x " n limn " = 1 "n n limn " a = 1 a > 0 " n n 2 n - 1 > 0 " " n n n = ( n - 1) + 1 (X, d)
(C) d(xn, xn+m) < . >0 n0"N n n0,m"N n
n
" n n n = ( n - 1)k k k=0 " " " n(n - 1) n n n n = 1 + n( n - 1) + ( n - 1)2 + . . . + ( n - 1)n .
2 >0 >0 " n(n - 1) n n - 1 > ( n - 1)2 2 " 2 n ( n - 1)2 < n
" 2 n 1 < n < 1 + , n n 2
" 2 n an = 1, xn = n, bn = 1 + , n = 2, 3, . . . n xn 1 b) 1 an = (1 + )n, n " N. n (an)
an+1 > an. n"N n n (a + b)n = an-kbk a, b " R n " N k=0 k
n
1 n 1 (1 + )n = ( )k = n k n k=0 1 n(n + 1) 1 n(n - 1)(n - 2) 1 1 + n + + + . . . n 2 n2 1 2 3 n3 n(n - 1) . . . (n - n + 1) 1 + = 1 2 . . . n nn 1 1 1 1 2 1 + 1 + (1 - ) + (1 - )(1 - ) + . . . + 2! n 3! n n 1 1 n - 1 (1 - ) . . . (1 - ) < n! n n 1 1 1 1 2 1 + 1 + (1 - ) + (1 - )(1 - ) + . . . 2! n + 1 3! n + 1 n + 1 1 1 n - 1 1 1 2 n + (1- ) . . . (1- )+ (1- )(1- ) . . . (1- ) = an+1. n! n + 1 n + 1 (n + 1)! n + 1 n + 1 n + 1 1 1 1 an = (1 + )n < 1 + 1 + + . . . + < n 2! n! 1 1 1 1 1 1 2 + + + . . . + = 2 + (1 + + . . . + ) = 2 22 2n-1 2 2 2n 1 1 - (1)n-1 2 2 + < 3. 1 2 1 - 2 e (an)
an = 0.
n0"N n n0 an+1 lim | | = ą < 1. n " an (an) z e (0) (z) = limn "(1 + )n, z " C n e H" 2, 71... (an) " R (an) + " - " ł ł
ł an > M an < młł . M"R n0"N n n0 m"R n0"N n n0 0 " " - " 0 " 1 " "0 00 0 " a) limn " Pk(n) P Q k l k Ql(n) l Pk(n) = asnk-s, Ql(n) = brxl-r, x " R. s=0 r=0 ńł 0 k < l, ł ł ł a0 ł Pk(n) k = l b0 lim = n " ł + " k > l '" a0 b0 < 0, Ql(n) ł ł ół - " k > l '" a0 b+0 < 0. b) limn "(7n - 5n - 3n - 1) n-n2 c) limn " 2n2-1 2n2+1 b) " - " [ "- "] lim (7n - 5n - 3n - 1) = lim [7n - (5n + 3n + 1)] = n " n " 5n 3n 1 lim 7n[1 - ( + + )] = [ " 1] = + ". n " 7 7 7n c) 1 " 1 limn " an = limn "(1 - )n = e-1 n n > 1 1 n - 1 n (n - 1) + 1 an = (1 - )n = ( )n = ( )-n = ( )-n = n n n - 1 n - 1 -n n-1 1 1 (1 + )-n = (1 + )n-1 . n - 1 n - 1 1 lim an = e-1 = . n " e b) n-n2 (2n2) + 1) - 2 lim xn = lim = n " n " 2n21 + 1 n-n2 ł łł n2 1 1 + n2+ 2 2 1 ł ł lim 1 - = (e-1)-1 = e. 1 n " n2 + 2 x0 " R A " R x0 A x0 A x0 f : Df R Df " R x0 Df f f g " R x0 limxx f(x) = g 0
lim xn = x0 ! lim f(xn) = g. n " n " (xn)"Sx0 )"Df f g " R x0
x " Sx , ! f(x) " Ug, . 0 Ug, Sx0, x"Df
0 < |x - x0| < ! |f(x) - g| < . >0 >0 x"Df sin x limx0 f(x) = 1 f(x) = x x " R \ {0} 1 1 "AOB sin x 2 x AOB 2 tgx "AOC 2 sin x x tgx Ą , x " (0, ). 2 2 2 2 sin x Ą Ą cos x 1, x " (- , 0) *" (0, ). x 2 2 Ą (xn) " (-Ą , 0) *" (0, n " N limn " xn = 0 2 2 sin xn cos xn 1, n " N. xn limn " f(xn) = 1 (xn) limx0 sin x = 1 x f : Df R g : Dg R D = Df )" Dg = "
D, Df, Dg " X (X, d) x0 D limxx f(x) = a limxx g(x) = b 0 0 1. limxx (f(x) ą g(x) = a ą b 0 2. limxx (f(x)g(x)) = a c 0 a 3. limxx f(x) = b = 0
0 g(x) b f : Df R Df " R
lim f(x) = g " R ! |f(x) - g| < . x+ " >0 >0 x>
lim f(x) = g " R ! |f(x) - g| < . x- " >0 <0 x< f : Df R Df " R x0 Df
(|x1 - x2| < ) '" (|f(x1) - f(x2)| ). >0 >0 x1,x2"R 1 x1 = n, x2 = n + > 0 n 1 n0 n n0 |x1 - x2| = < |f(x1) - n f(x2)| = |n2 - (n + 1)2| = |2n - 1| 1 = 1 c) f(x) = , x " (0, 1] x1 = x 1 2 1 , x2 = |x1-x2| = < n |f(x1)-f(x2)| = n n n n 1 1 |n - | = n = 2 2 2 f : [a, b] R [a, b] f : Ux R 0 f x0 A " R : U0 R
f(x) - f(x0) = A (x - x0) + (x - x0)
x"Ux0 "x "f lim (x - x0) = 0, xx0 "x = x - x0 "f = f(x) - f(x0) "x A f x0 A = f (x0) A"x f x0 A (x - x0) = df(xo, "x) f(x) - f(x0) = A + ("x), "x f(x) - f(x0) f (x0) = lim . "x0 - x0 x a) f(x) = x2 x0 = 1 f x0 = 1 "x = x - x0 = x - 1 "f "f f(x) - f(1) x2 - 1 (x0) = (1) = = "x "x x - 1 x - 1 "f "f lim (x0) = lim (1) = "x0 x1 "x "x x2 - 1 lim = lim(x + 1) = 2. x1 x1 x - 1 f (1) - 2 b) f(x) = x2, x " R Df = R (x2) = 2x, x " R. x0 " R Df f(x0 + "x) - f(x0) lim (x0) = lim = "x0 "x0 "x "x (x0 + "x)2 - x2 (x0 + "x - x0)(x0 + "x + x0) 0 lim = lim = "x0 "x0 "x "x "x(2x0 + "x) lim = lim (2x0 + "x) = 2x0. "x0 "x0 "x x0 f (x) = (x2) = 2x x " R f, g x0 " R f f ą g fg f = 0 x0
g 1) (f ą g) = f ą g 2) (fg) = f g + fg 3) (f) = f f g-fg 4) (f ) = g g2 f(x) = signx x2, x " R ńł ł -2x jeżeli x < 0 ł f (0) = 0 jeżeli x = 0 ł ół 2x jeżeli x > 0 (x) = signx, x " R (x) = x2, x " R f f(x) = (x) (x), x " R x " (- ", 0) *" (0, + ") f (x) = (x) (x) + (x) (x) =
(-1) 2x jeżeli x < 0 1 2x jeżeli x > 0 f x0 = 0 "f sign(0 + "x) (0 + "x)2 lim (0) = lim = "x0 "x0 "x "x (sign"x) ("x2) lim = lim sign"x "x = 0. "x0 "x0 "x
f (0) = 0 f : (a, b) Y (a, b) Y
f (x) > 0 (" f (x) < 0. x"(a,b) x"(a,b) Y y " Y 1 (f-1) (y) = , y = f(x) ! x = f-1(y). f (x) 1 " arcsin (x) = , x " (-1, 1). 1 - x2 Ą f = sin (-Ą , ) 2 2 Ą Ą f (x) = sin (x) = cos x > 0, x " (- , ). 2 2 1 1 1 1 " " (f-1) (x) = = = = . cos y cos arcsinx 1 - x2 1 - sin2 arcsinx f : Ux R g : Uf(x ) R f(Ux ) " Uf(x ) 0 0 0 0 f X0 g u0 = f(x0) g ć% f x0 (g ć% f) (x0) = g (f(x0))f (x0). f(x) > 0 x " Df 1 (ln f(x)) = f (x). f(x)
f (x) h (x) = eg(x) ln f(x) g (x) ln f(x) + g(x) , x " Df. f(x) f : Ux R f(1) Ux 0 0 f(1) x0 2 x0 f(2) n f f(n) = (f(n-1)) , n " N. K f (x0, f(x0)) y - f(x0) = f (x0)(x - x0). K (x0, f(x0)) K f (x0, f(x0)) 1 y - f(x0) = - (x - x0) f (x0) = 0
f0(x0) x = x0 f (x0) = 0 f : Df R f K = {(x, f(x)) : x " Df} K K :
y = f(x) x " Df K f P = (x0, f(x0) K P f x0 s = [1, f (x0)] s P K P f : [a, b] R [a, b] (a, b) f(a) = f(b) " (a, b) f () = 0 (, f()) K = {(x, f(x)) : x " [a, b]} " (a, b) Ox f : [a, b] R [a, b] " (a, b) f(b) - f(a) = f (). b - a f : [a, b] R a, b " R# [a, b] L 0
|f(x1) - f(x2)| L |x1 - x2|. x1,x2"[a,b] f : [a, b] R (a, b) f L L = sup{|f (x)| : x " (a, b)}. x1, x2 " (a, b) x1 < x2 [x1, x2] " (x, y) f(x1) - f(x2) f () = . x1 - x2 |f(x1) - f(x2)| |f ()| |x1 - x2| L |x1 - x2|. a) f(x) = sin x, x " R |f(x1) - f(x2)| = | sin x1 - sin x2| = x1 - x2 x1 + x2 |2 sin cos | 2 2 x1 + x2 x1 - x2 2 | cos | | sin | 2 2 |x1 - x2| 2 1 = |x1 - x2|. 2 f (x) = cos x, x " R L = sup{|f (x)| : x " R} = 1 b) f(x) = xą x 0 ą " (0, 1) [0, ") [0, ") f : [a, b] R (a, b) f (x) > 0, x " (a, b) f (x) < 0, x " (a, b) f (a, b) f g x0 " int(Df) lim f(x) = 0 = lim g(x) xx+ xx+ 0 0 lim f(x) = " = lim g(x), xx+ xx+ 0 0
lim f(x) = 0 = lim g(x) xx- xx- 0 0
lim f(x) = " = lim g(x) xx+0- xx- 0 R#
f (x) f (x) lim lim , g (x) xx+ xx- g (x) 0 0
f(x) f(x) lim lim g(x) xx+ xx- g(x) 0 0
f(x) f (x) f(x) f (x) lim = lim lim = lim . g(x) g (x) xx- g (x) xx+ xx+ xx- g(x) 0 0 0 0 1) limx0 arcsinx sin x 1 " arcsinx 0 H 1-x2 lim = [ ] = lim = 1. x0 x0 sin x 0 cos x 2) limx " x1000 ex x1000 " 1000x999 " H H lim = [ = lim = [ ] = x " x " ex " ex " 1000! 1000! . . . = lim = [ ] = 0. x " ex " x2 2 3) limx0 Ą arctgx 2 x x2 2 2 Ą lim arctgx = [00] = lim eln( arctgx) = x0 x0 Ą 2 2 Ą lim ex ln ( arctgx) x0
ł f(x) < f(x0) f(x) > f(x0)łł . x"Sx0 x"Sx0 f : Ux R x0 0 f (x0) = 0 f(x) = |x| x " R x0 = 0 f (0) x0 an + bn = cn n 2 a, b, c " Z f :
Ux R Sx x0 - f (x) > 0 0 x"Sx0
0 '" + f (x) < 0, f x0 x"Sx0
- f (x) < 0 '" + f (x) > 0, x0 x"Sx0 x"Sx0 f : Ux R ) Cn(Ux ) n = 2k k " N 0 f (x0) = f (x0) = . . . = f(n-1)(x0) = 0, f(n)(x0) = 0 f x0
f(n)(x0) < 0 f(n)(x0) > 0 f(x) = |x| x " R x0 = 0
-1 x < 0 f f (x) = 1 x > 0 x0 = 0 f(x) = ex + e-x + 2 cos x f (x) = ex - e-x - 2 sin x '" f (x) = 0 ! x = 0, f (x) = ex + e-x - 2 cos x '" f (0) = 0, f (x) = ex - e-x + 2 sin x '" f (0) = 0, f(4)(x) = ex + e-x + 2 cos x '" f(4)(0) = 4 > 0. f x0 = 0 R A " R (xn) " A (xn ) A k limk " xn = x " A k a) [0, 1] R b) A = [0, 1] \ Q R A " R
K0(x, r) " A x"A K0(x,r) A " R R R \ A A " R R
[ ! x " A]. (xn)"A limn " xn=x (a, b) R [a, b] {a}, {b}, . . . " R A " R R R A f : A R A R A " A f() = inf{f(x) : x " A} " A f() = sup{f(x) : x " A}. f : A R f Int(A) f A A f " A f() = 0 f f f A
a) f(x) = (1 - x2)(1 + 2x) [-1, 1] f(-1) = f(1) = 0 f x(1 - 4x2)
f (x) = . (1 - x2)(1 + 2x2) 1 0, -1, 2 2 " 1 3 2 1 f(1) = f(-1) = 0, f(0) = 1, f(- ) = = f( ). 2 4 2 f 0 f(-1) = f(1) = 0 " 3 2 2 2 4 " f(-1) = f(1) = " b) f(x) = x2 - 3x + 2 + 3x - x2, x " [0, 1] *" [2, 3] f " f(0) = f(1) = f(2) = f(3) = 2. f 2x - 3 3 - 2x " " f (x) = + . 2 x2 - 3x + 2 2 3x - x2 " " 3 3- 5 3+ 5 3 f (x) = 0 ! x = (" x = (" x = " Df 2 2 2 2 " " 3 - 5 3 + 5 3 f( ) = f( ) = . 2 2 2 " f 1, 2, 3, 4 2 " " 3- 5 3+ 5 3 2 2 2 f : (a, b) R f (a, b) (a, b) x0 " (a, b) Sx x0 (x0, f(x0)) 0 f f (a, b) f : (a, b) R x0 " (a, b) x0 x0 f f(x) = signx x2 x " R
-x2 x 0 f(x) = x2 x > 0 ńł ł -2x jeżeli x < 0 ł f (x) = 0 jeżeli x = 0 ł ół 2x jeżeli x > 0 ńł ł -2 jeżeli x < 0 ł f (x) = jeżeli x = 0 ł ół 2 jeżeli x > 0 f (- ", 0) (0, ") x0 = 0 f f x0 x = x0 f
lim f(x) " {- ", + "} lim f(x) " {- ", + "} . xx- xx+ 0 0 x = x0 f f x = 0 f(x) = xn x " R
- " jeżeli n = 1, 3, . . . lim f(x) = + " jeżeli n = 2, 4, . . . x0- lim f(x) = + " n " N. x0+ f + " - " f : (- ", a) *" (b, ") R x f(x)
f(x) f(x) lim = m " R lim = m x- " x+ " x x
lim (f(x) - mx) = k " R lim (f(x) - mx) = k , x- " x+ " y = mx + k f k " R m = 0 1 Df f(x) = x ln (e + ) x x " Df f Df f 1 1 x = 0 '" e + > 0 ! x " (- ", - ) *" (0, ")
x e ex + 1 1 1 x = 0 '" > 0 ! x = 0 '" e(x + )x > 0 ! x " (- ", - ) *" (0, ")
x e e f(x) 1 lim = lim ln (e + ) = 1 = m. x- " x- " x x 1 1 lim (f(x) - mx) = lim (x ln (e + ) - x) = lim x(ln (e + ) - 1) = [(- ") 0]. x- " x- " x- " x x [ " 0] 1 ln (e + ) - 1 0 H x lim (f(x) - x) = lim = [ ] = 1 x- " x- " 0 x 1 1 (- ) 1 x2 1 1 e+ x lim = lim = = k. 1 1 x- " x- " - e + e x2 x 1 y = x + f e x = -1 e 1 1 lim x ln (e + ) = [(- ) (- ")] = + ". - 1 x e x- e x = -1 e f x = 0 f 1 lim f(x) = lim x ln (e + ) = [0 "] = x)+ x0+ x 1 ln (e + ) " H x lim = [ ] = x0+ 1 " x 1 1 (- ) 1 x2 1 1 e+ x lim = lim = [ ] = 0. x0+ 1 x0+ 1 (- ) e + " x2 x x = 0 + " f(x) 1 lim = lim ln (e + ) = 1 = m x+ " x+ " x x 1 lim (f(x) - mx) = lim x ln (e + ) - x) = x+ " x+ " x 1 lim x[ln (e + ) - 1] = [ " 0] = x+ " x 1 ln (e + ) - 1 0 H x lim = [ ] = 1 x+ " 0 x 1 1 (- ) 1 x2 e+ x lim = 1 x+ " - x2 1 1 lim = = k. 1 x+ " e + e x 1 y = x + f e f f : {1, . . . , n} R t ft t t n f t ft+1 - ft , t = 1, 2, . . . , n. ft f n t+"t f (0, ") R t f(t) f f(t + "t) - f(t) lim "t0 "t f(t) f(t + "t) - f(t) f (t) lim = . "t0 "t f(t) f(t) f x f : (0, ") (0, ") x f(x) (0, ") x0 "x "f f(x0 + "x) - f(x0) df (x0) = "x "x "x x0 "x 0 "f f(x0 + "x) - f(x0) df (x0) = = f (x0). "x "x x f (x) "f = f(x0 + "x) - f(x0) H" f (x0)"x. "x f (x0) "x f : (0, ") (0, ") x f(x) (0, ") x0 " (0, ") "x > 0 f(x0 + "x) - f(x0) "x : f(x0) x0 f [x0, x0 + "x] E f(x0 + "x) - f(x0) x0 x0 lim = f (x)) "x0 "x f(x0) f(x0) f x0 E(x0) f p% q% q H" pE(x) f : Ux R Ux x0 " R 0 0 f Cn Ux n " N 0 f n f Cn(Ux ) f " Cn(Ux ) C0(Ux ) 0 0 0 Ux ) C "(Ux ) n 0 0 n " N Cn(A) A R C2((0, 1)) (0, 1) f Cn-1 [a, b] n " (a, b) n-1
In(x) = sinn xdx = sinn-2 x sin2 xdx = sinn-2 x(1 - cos2 x)dx =
sinn-2 xdx + Jn(x),
Jn(x) = sinn-2 x cos2 xdx = cos x[sinn-2 cos x]dx =
f(x) = cos x g (x) = sinn-2 x cos x
= sinn-1 x
f (x) = - sin x g(x) = n-1 1 1 cos x sinn-1 x + In(x). n - 1 n - 1 n - 1 1 1 In(x) = In-2(x) - ( cos x sinn-1 x + In(x)), n n - 1 n - 1 n - 1 1 In(x) = In-2(x) - sinn-1 x cos x. n n
sin4 xdx 3 1 I4(x) = I2(x) - sin3 x cos x. 4 4 1 1 1 1 I2(x) = I0(x) - sin x cos x = x - sin x cos x. 2 2 2 2
3 1 1 1 sin4 xdx = ( x - sin x cos x) - sin3 x cos x. 4 2 2 4 Pn(x) f(x) = x " Df Qn(x) n < m f A 1. f(x) = a = 0
ax+b A 2. f(x) = a = 0
ax2+bx+c Ax+B 3. f(x) = a = 0
ax2+bx+c
Adx dx A = A = ln |ax + b| + C. ax + b ax + b a ą " = b2 - 4ac = 0 ax2 + bx + c = a(x - p)2
dx 1 dx -1 = = + C. ax2 + bx + c a (x - p)2 a(x - p) " < 0 b " ax2 + bx + c = a(x - p)2 + q, p = - , q = - . 2a 4a " ax2 + bx + c = a[(x - p)2 - ]. 4a2
dx 1 dx = = " ax2 + bx + c a - p)2 + (- ) (x 4a2
1 dx " , A = - . a (x - p)2 + A2 4a2
dx 1 1 x - p = arctg( ) + C = ax2 + bx + c a A A 1 1 x - p
dx 1 dx = . 3x2 - 9x + 6 3 (x - 1)(x - 2) 1 A B = + . (x - 1)(x - 2) x - 1 x - 2 1 = A(x - 2) + B(x - 1). x = 2 ! B = 1 x = 1 ! A = -1
1 -1 1 1 x - 1 I = [ + ]dx = ln | | + C. 3 x - 2 x - 1 3 x - 2 mx+k 3. f(x) = ax2+bx+c f
f (x) dx f(x) ax2+bx+c
k x + m I = m dx = ax2 + bx + c
2ak m 2ax + m dx = 2a ax2 + bx + c
m (2ax + b) + (2ak - b) m dx = 2a ax2 + bx + c
m 2ax + b m 2ak dx dx + ( - b) = 2a ax2 + bx + c 2a m ax2 + bx + c
m 2ak - bm dx ln |ax2 + bx + c| + . 2a 2 ax2 + bx + c
xdx x2+x+1
1 2xdx 1 2x + 1 - 1 I = = dx = 2 x2 + x + 1 2 x2 + x + 1
1 2x + 1 1 dx dx - = 2 x2 + x + 1 2 x2 + x + 1 " " 1 1 2 3 2 3 1 ln(x2 + x + 1) - arctg( (x + )) + C. 2 2 3 3 2 f : I R I = [a, b] n I [x0, x1], [x1, x2], . . . , [xn-1, xn], a = x0 < x1 < . . . < xn = b, n " N i i = xi - xi-1 i [xi-1, xi] i = 1, . . . , n n "n = max{i : 1 i n}. [xi-1, xi] i xi-1 i xi i " {1, . . . , n} n
n = f(i)i. i=1 ( n)n"N I ( n)n"N I lim "n = 0. n " ( n) I i i " {1, . . . , n} n " N (n) Ł n
Ł = lim f()i, n " i=1 f R I Ł f [a, b]
b Ł = f(x)dx. a mi = inf{f(x) : x " [xi-1, xi]}, Mi = sup{f(x) : x " [xi-1, xi]}, n
sn = mii i=1 n
Sn = Mii. i=1 sn Rn Sn, n " N, n [a, b]
b f(x)dx a lim (Sn - sn) = 0 n " ( n) R R f [a, b] ( n) [a, b] (sn) (Sn) [xi-1, xi] f [xi-1, xi] i " [xi-1, xi] f(i) " [mi, Mi] n n n
sn = mii f(i)i Mii = Sn i=1 i=1 i=1 mi Mi xi-1, xi] f [a, b] > 0 > 0 |x - x | < |f(x ) - f(x )| < > 0 > 0 i < n
i = 1, . . . , n |Mi - mi| < |Mi - mi|i < Sn - sn < i=1 b-a limn "(Sn - sn) = 0
1 x " Q f(x) = 0 x " R \ Q R S = b - a s = 0 A B A B f : A B A A B " A B = A f : A B
card(") = 0. card({1}) = card({a}) = 1, . . . card({1, 2, . . . , n}) = n, . . . . 5!0 card(N) = 5!0. 5!0 card(Z) = card(Q) = 5!0. A A (an)n"N A = {a1, a2, . . .}. card(R) = c. a b A a B b C C " B card(A) = card(C) a b card(A) = card(B)
A B a < b 5!0 < c A card(A) < card(P(A)). P(A) A A card(N) = 5!0 card(P)(N) = c A " R > 0 A (ai, bi) i " N "
(ai, bi) " A i=1 "
|bi - ai| < . i=1 R f : I R R I = [a, b] f f [a, b] [a, b] R [a, b] R [a, b] f f f R [a, b] |f| R f R f R [a, b] " R
b b f(x)dx = f(x)dx. a a f g R [a, b] R
b b b (f(x) + g(x))dx = f(x)dx + g(x)dx. a a a f R [a, b] c " (a, b)
c b f(x)dx = f(x)dx + f(x)dx. a c f R [a, b] f(x) 0 x " [a, b]
b f(x)dx 0. a f g R [a, b] f(x) g(x) x " [a, b]
b b f(x)dx g(x)dx. a a
b b a f(x)dx f(x)dx = - f(x)dx a a b R [a, b]
b 1 m f(x)dx M. b - a a "
3 I = 3 + x3dx 1 " m = 2, M = 30, b - a = 2. " 2 2 I 30 2. " I " [4, 2 30] f [a, b] " [a, b]
b f(x)dx = (b - a) f(). a f R [a, b]
x Ś(x) = f(t)dt, x " [a, b]. a f R [a, b] a) Ś [a, b] b) f x = t Ś f(x) Ś (x) = f(x)
x3 Ś(x) = cos t2dt. 1 x 1 x3 x Ś(x) = - cos t2dt + cos t2dt. 1 1 Ś 1 1 Ś (x) = - cos x2(- ) + cos x6(3x2) = cos x2 + 3x2 cos x6. x2 x2 1. f : [a, b] R [a, b] 2. : [ą, ] [a, b] a) (ą) = a () = b b) C1 [ą, ]
b f(x)dx = f((t)) (t)dt. a ą 2 1
sin dx Ą x I = 1 x2 Ą ńł ł
ł x t ł ł 1 żł = t
1 x I = Ą = 1 Ą ł - dx = dt ł ół ł 2 Ą x2
Ą 2 Ą Ą 2 - sin tdt = sin tdt = Ą Ą 2 Ą Ą - cos t|Ą = - cos Ą + cos = -(-1) + 0 = 1. 2 2 f g C1 [a, b]
b b f(x)g (x)dx = [f(x) g(x)]|x=b - f (x)g(x0dx. x=a a a
1 I = x arctgxdx 0
f(x) = arctgx g (x) = x I = = 1 x2 f (x) = g(x) = 1+x2 2
1 1 0 x x 1 e e (Dx)1 : (Dx)2 : -1 y x - 1 ln x y x - 1. D (Dx)1 (Dx)2 Ox 1 1 e |D| = |(Dx)1 + |(Dx)2| = [(x - 1) - (-1)]dx + [(x - 1) - ln x]dx. 1 0 e D Oy
-1 x 0 Dy : y + 1 x ey
0 y2 e - 2 |D| = |Dy| = [ey - y - 1]dy = [ey - - y]|0 = . . . = . -1 -1 2 2e R f : D R D (- ", a] - " [a, + ") + " D = (- ", + ") D = (- ", a] a " R f R Dą = [ą, a] " (- ", a]
A " 0+ 1 " " 3 3 lim [3 1 - 3 ] + lim [-e1-A + 1] = 3 + 1 = 4. A " 0+ X + : X X X (x, y) x + y R X X (, x) x. X 1) x + y = y + x x, y " X 2) x + (y + z) = (x + y) + z x, y, z " X 3 Ś " X
x + Ś = Ś = x = x. x"X 4) x " X -x (-x) + x = x + (-x) = Ś. 5) (x + y) = x + y " R, x " X 6) ( + ) x = x + y , " R x " X 7) ( x) = () x , " R, x " X 8) 1 x = x x " X < X, +, , R > < X, +, , R > (-x) x " X x " X -(-x) = x, 0 x = Ś, -x = (-1) x, x + . . . = x = n x, n " N.
n a) R =< R, +, , R > b) X = R2 =< R R, +, , R > x = (x1, x2) y = (y1, y2) x + y = (x1 + y1, x2 + y2) " R x = (x1, x2). c) X = R3 x = (x1, x2, x3) y = (y1, y2, y3) x + y = (x1 + y1, x2 + y2, x3 + y3) x = (x1, x2, x3) = (x1, x2, x3). c) Rn x = (x1, . . . , xn) y = (y1, y2, . . . , yn) x + y = (x1 + y + 1, x2 + y2, . . . , xn + yn), x = (x1, x2, . . . , xn), " R. Rn x1, x2, . . . , xm " X < X, +, , R > 1, 2, . . . , m 2 + 2 + . . . + 2 > 0 1 2 m m
+ = 0 + 2 = 0 = = 0 2 + 2 > 0 x x1, . . . , xm 1, . . . , m m
x = lxl = 1 x1 + . . . + m xm. l=1 a) x = (1, 1), y = (3, 3) R2 y = 3 x b) R3 a = (1, 0, 1), b = (0, 1, 0), c = (1, 1, 1) c = a + b < X, +, , R > W " X W X + W W
1 0 -1 - A w : R R wA() w(t) = a0tn + a1tn-1 + . . . + = an, t " R w(A) = a0 An + a1 An-1 + . . . + an I, t " R, A, I " Matn nn A " Matn wA() = 0, wA(A) = Ś Ś n n a0 An + a1 An-1 + . . . + an-1 A + an I = Ś. (") an I = -a0 An - a1 An-1 - . . . - an-1 A. A A-1 (") A-1 an A-1 " I = -a0 A-1 " An + . . . - an-1 A-1 " A -a0 -an-1 A-1 = An-1 + . . . I. an an
2 3 A = -2 5 2 - 7 + 16 = 0. A2 - 7 A + 16 I = Ś. 7 1 16 I = -A2 + 7 A ! A-1 = I - A. 16 16
xi 0. i"{1,...,n} B (1) (1 ) A " X = Ś (3) A " X = B, A " Matn = Matn n X, B " Matn1 (3) (3) A " X = B ! A-1 " (A " X) = A-1 " B ! (4) X = A-1 " B. (4) Wj xj = , i = 1, . . . , n, W
R(A) = R(U) = n R(A) = R(U) = r < n p = n - r R(A) = R(U) = n (1) m n m = n (1) U
U a" A B , A ńł ł a x1 + a x2 + . . . + a xn = b ł 11 12 1 n 1 ł ł ł a x2 + . . . + a xn = b 22 2 n 2 ł ł ł ł ół a xn = b n n n b n xn = a n n xn-1 n x1
ńł 1 3 ł x = - ą, ł 2 2 1 y = -1 - ą, 2 2 ł ół z = ą, ą " R. R2 P = (xP , yP ) Q = (xQ, yQ) R2 d : R2 [0, ")
. df (P, Q) = ((xP , yP ), (xQ, yQ)) d(P, Q) = (xP - xQ)2 + (yP - yQ)2 P, Q, R " R2 d(P, Q) = 0 ! P = Q d(P, Q) = d(Q, P ) d(P, Q) d(P, R) + d(R, Q) R2 d R2 P0 = (x0, y0) " R2 r P0 r K0(P0, r) = {P = (x, y) " R2 : d(P0, P ) < r}. P0 r K-(P0, r) = {P " R2 : d(P, P0) r}. A " R2 K0(P0, r) A " K0(P0, r) A " R2 P0 " R2 A K0(P0, r) K0(P0, r) " A A A0 Int(A) A A R2 A A A0 = A a) R2 b) R2 I0 = (a, b) (c, d) c) " d) R2 UP P0 " R 0 P0 UP = K0(P0, r) 0 SP SP = UP \ {P0}. 0 0 0 P0 " R2 A " R2 P0 A A R2 R2 A R2 A = R2 \ A R2 (Pn) Pn = (xn, yn) n " N P = (x, y)
d((xn, yn), (x, y)) < . >0 n n0 n0"N limn " Pn = P Pn = P limn " Pn = P P P lim Pn = P ! lim xn = x '" lim yn = y. n " n " n " A " R2 A Ad A " R2 A R2 A-
P " A- ! [n " Pn = P ! P " A-]. lim (Pn)"A A R2 "A = A- \ A0. A = K0(P0, r) *" {P1} P1 d(P1, P ) > r + 1 K-(P0, r) A {P " R2 : d(P, P0) = r}. P1 A A A Rn n = 1, 2, . . . d : Rn Rn [0, ")
df n (x, y) d(x, y) = |xi - yi|2, k=1 x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn) Rn R R2 f : Df R " = Df " R2
f : Df R . P = (x, y) f(P ) = f(x, y) f f 2 a) f(x, y) = x2 + y2 (x, y) " R2 f Df = R2 z = x2 + y2 " b) f(x, y) = 1 - x2 - y2 (x, y) " Df Df = {(x, y) " R2 : x2 + y2 1} = K-((0, 0), 1). " z = 1 - x2 - y2 z 0 ln (x2+y2-1) " c) f(x, y) = f 4-x2-y2 Df = {(x, y) " R2 : 1 < x2 + y2 < 4}. f lim f(x, y) = + ", lim f(x, y) = - ". x2+y24- x2+y21+ f : Df R P0 Df Df " R2 f g " R# (Pn)n"N " SP )" Df 0 limn " Pn = P0 limn " f(Pn) = g. f : Df R Df " R2 P0 " Df P0 Df f P0 P0 " Df P0 f P0 (Pn)n"N " Df limn " Pn = P0 limn " f(Pn) = f(P0). R2 f : Df R Df " R2 A " Df f A gć%f f : Df R Df " R2 g : Dg R f(Df) " Dg Df f : Df R Df R2 1) Df 2) (0, 0) " Df f(0, 0) = max{f(P ) : P " Df} (1, 1) " Df f(1, 1) = min{f(P ) : P " Df} n n 3 A " Rn f f : Df R Df " R2 P0 = (x0, y0) Df = [ą, ] a f(x0 + ąt, y0 + t) - f(x0, y0) lim , t0 t f (x0, y0) v "f (x0, y0) " v f (x0, y0)
x = x0 + ąt, (l) y = y0 + t, t " R, Ą OY f (t) t t0 = 0 "f (x0, y0) " v
v f : UP R P0 = (x0, y0) 0 "f f (x0, y0) "x "f x vx = [1, 0]
"vv
"f "f (x0, y0) = (x0, y0) = "x "vx
f(x0 + "x, y0) - f(x0, y0) lim . "x0 "x "f f (x0, y0) "y "f y vy = [0, 1]
"vy
"f "f (x0, y0) = (x0, y0) = "y "vy
f(x0, y0 + "y) - f(x0, y0) lim . "y0 "y n n 3 0 f : Ux R x0 = (x0, . . . x0 , x0, x0 , . . . , x0) 1 i-1 i i+1 n "f (x0, . . . x0, x0 , . . . , x0) = "xi 1 i i+1 n lim "xi0 f(x0, . . . , x0 , x0 + "xi, x0 , . . . , x0) - f(x0, . . . , x0 , x0, x0 , . . . , x0) 1 i-1 i i+1 n 1 i-1 i i+1 n . "xi f : Ux0 R Ux0 " R2 f x0 A, B " R f(x0 + "x1, x0 + "x2) - f(x0, x0) = A"x1 + B"x2 + ("x1, "x2), 1 2 1 2 : U(0,0) R ą) (0, 0) = 0 ("x1,"x2) ) lim("x ,"x2)(0,0) " = 0 1 "2x1+"2x2 a) f x0 "f "f "f x0 A = (x0) "x "y "x "f B = (x0) "y b) f x0 f x0 f (x0, y0) "f "f df df(x0, y0) = (x0, y0)dx + (x0, y0)dy "x "y f (x0, y0) f "f x0 "x "2f (x0) x0 ptx2 "f "f "2f (x0 + "x, y0) - (x0, y0) "x "x (x0, y0) = lim . "x0 "x2 "x "2f "2f "2f (x0, y0) (x0, y0) (x0, y0) "x"y "y"x "y2 f "3f "3f "3f "3f (x0, y0), (x0, y0), (x0, y0), (x0, y0), "x3 "x"y2 "y2"x "x2"y "3f "3f (x0, y0), (x0, y0). "y"x2 "y3 4 4 "2f "2f "x"y "y"x (x0, y0) f (x0, y0) "2f "2f "2f df d2f(x0, y0) = (x0, y0)dx2 + 2 (x0, y0)dxdy + (x0, y0)dy2. "x2 "x"y "y2 m f (x0, y0) m f
f (0, 0) n 1 f(x1, . . . , xn) = (xi + ) i=1 xi "f 1 "f = 1 - i = 1, . . . , n = 0, i = 1, . . . , n "xi x2 "xi i 2n P = (1, 1, . . . , 1) Q = (-1, -1, . . . , -1) -1 2n - 2 1 -1 "2f 2 "2f = i = 1, . . . , n = 0 i = j i, j =
"x2 x3 "xixj i i 1, . . . , n P ł łł 1 0 . . . 0 0 ł śł ł śł 0 1 . . . 0 0 ł śł ł śł A = ł śł . ł śł ł śł 0 0 . . . 1 0 ł ł 0 0 . . . 0 1 Mi = 1 > 0 i = 1, . . . , n P f(P ) = 2 Q ł łł -1 0 . . . 0 0 ł śł ł -1 . . . 0 0 śł 0 ł śł ł śł A = ł śł . ł śł ł śł 0 0 . . . -1 0 ł ł 0 0 . . . 0 -1 (-1)iMi = 1 > 0 i = 1, . . . , n Q f(Q) = -2 2n-2 f : Df R Df " Rn Df f f Df f Df "Df Df A Rn A Rn f Int(Df) f Df "Df Df n - 1, n - 2, . . . , 1, 0 A. "Df Df f(x, y) = sin x + sin y - sin (x + y) Df = {(x, y) " R2 : x + y 2Ą '" x 0 '" y 0} "f "f = cos x - cos (x + y) = cos y - cos (x + y) "x "y
cos x - cos (x + y) = 0 cos x = cos y ! ! cos y - cos (x + y) = 0 cos y = cos (x + y) (x = y + 2kĄ (" x = -y + 2kĄ) '" (cos y = cos (x + y)). x = y cos x = cos 2x 2 Df (2Ą, Ą) 3 3 " " " " 2 2 4 3 3 3 3 f(2Ą, Ą) = sin Ą - sin Ą = + + = 3 3 3 3 3 2 2 2 2 A = {(x, 0) : 0 x 2Ą} f(x, 0) = sin x - sin x = 0 B = {(0, y) : 0 y 2Ą} f(0, y) = 0 C = {(x, 2Ą - x) : 0 x 2Ą} f(x, 2Ą - x) = sin x + sin (2Ą - x) - sin 2Ą = 0 2 3 " Df (2Ą, 3Ą) 3 3 0 2 Df f f(x, y) = xy(1 - x - y) f(x, y) = xy - x2y - xy2 "f "f (x, y) = y - 2xy - y2, (x, y) = x - x2 - 2xy. "x "y
"f (x, y) = 0 y(1 - 2x - y) = 0 "x ! "f (x, y) = 0 x(1 - x - 2y) = 0 "y x = 0 y = 0 y = 1 (0.0), (0.1) y = 0 x = 0 x = 1 (1, 0) 2xy = 2xy y - y2 = x - x2 y - x = y2 - x2 y - x = y2 - x2 ! (y - x)(1 - x - y) = 0. 1 x = y x2 - 3x2 = 0 (1, ) 3 3 "2f "2f "2f (x, y) = -2y, = -2x, (x, y) = 1 - 2x - 2y. "x2 "y2 "x"y
0 1 (0, 0). 1 0
0 -1 (1, 0). -1 -2
-2 -1 (0, 1). -1 0
1 1 -2 -1 3 3 ( , ). -1 -2 3 3 3 3 1 (1, ) 3 3 1 1 f(1, ) = 3 3 27 (0, 0) (1, 0) (0, 1) f (1.0) f x - y = 1 = [1, 1] v y = x - 1 (x) = f(x, x - 1) = -2x2(x - 1) x > 1 x < 1 (0, 1) f y = x R R A-1 (1)